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Reflecting Brownian Motion and the Gauss–Bonnet–Chern Theorem


We use reflecting Brownian motion (RBM) to prove the well-known Gauss–Bonnet–Chern theorem for a compact Riemannian manifold with boundary. The boundary integrand is obtained by carefully analyzing the asymptotic behavior of the boundary local time of RBM for small times.

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The second author would like to thank Professor Denis Bell of the University of North Florida for his interest in the project and helpful discussions during the early stage of the investigation. The authors also thank Professor Xiaonan Ma of Universite Paris 7 for drawing their attention to the papers [3] and [4].

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Correspondence to Weitao Du.

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Du, W., Hsu, E.P. Reflecting Brownian Motion and the Gauss–Bonnet–Chern Theorem. Commun. Math. Stat. (2022).

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  • Manifold with boundary
  • Gauss–Bonnet–Chern theorem
  • Reflecting Brownian motion

Mathematics Subject Classification

  • 60J65
  • 58J65